If the length of direct common tangent and transverse common tangent of two circles with integral radii are 3 units and 1 unit respectively, then the reciprocal of the square of the distance between the centres of the circles is equal to

Answer

To solve the problem, we need to find the reciprocal of the square of the distance between the centers of two circles given the lengths of their direct and transverse common tangents.

### Step-by-Step Solution:

1. **Understanding the Problem:**
We have two circles with radii \( R_1 \) and \( R_2 \). The lengths of the direct common tangent (DCT) and the transverse common tangent (TCT) are given as 3 units and 1 unit respectively.

2. **Using the Formulas:**
The formulas for the lengths of the direct common tangent (DCT) and the transverse common tangent (TCT) between two circles are:
- For DCT:
\[ L_{DCT} = \sqrt{d^2 - (R_1 + R_2)^2} \]
- For TCT:
\[ L_{TCT} = \sqrt{d^2 - (R_1 - R_2)^2} \]
where \( d \) is the distance between the centers of the circles.

3. **Setting Up the Equations:**
Given:
- \( L_{DCT} = 3 \)
- \( L_{TCT} = 1 \)

We can set up the equations:
\[ \sqrt{d^2 - (R_1 + R_2)^2} = 3 \quad \text{(1)} \]
\[ \sqrt{d^2 - (R_1 - R_2)^2} = 1 \quad \text{(2)} \]

4. **Squaring Both Equations:**
Squaring equation (1):
\[ d^2 - (R_1 + R_2)^2 = 9 \quad \Rightarrow \quad d^2 = 9 + (R_1 + R_2)^2 \quad \text{(3)} \]

Squaring equation (2):
\[ d^2 - (R_1 - R_2)^2 = 1 \quad \Rightarrow \quad d^2 = 1 + (R_1 - R_2)^2 \quad \text{(4)} \]

5. **Equating the Two Expressions for \( d^2 \):**
From equations (3) and (4):
\[ 9 + (R_1 + R_2)^2 = 1 + (R_1 - R_2)^2 \]

6. **Simplifying the Equation:**
Rearranging gives:
\[ (R_1 + R_2)^2 - (R_1 - R_2)^2 = 1 - 9 \]
\[ (R_1 + R_2 - (R_1 - R_2))(R_1 + R_2 + (R_1 - R_2)) = -8 \]
\[ (2R_2)(2R_1) = -8 \quad \Rightarrow \quad 4R_1R_2 = -8 \quad \Rightarrow \quad R_1R_2 = 2 \]

7. **Finding Possible Values for \( R_1 \) and \( R_2 \):**
Since \( R_1 \) and \( R_2 \) are integral radii, the possible pairs \((R_1, R_2)\) are \((2, 1)\) or \((1, 2)\).

8. **Substituting Back to Find \( d^2 \):**
Using \( R_1 = 2 \) and \( R_2 = 1 \):
\[ d^2 = 9 + (2 + 1)^2 = 9 + 9 = 18 \]

9. **Finding the Reciprocal of the Square of the Distance:**
The reciprocal of the square of the distance between the centers is:
\[ \frac{1}{d^2} = \frac{1}{18} \]

### Final Answer:
The reciprocal of the square of the distance between the centers of the circles is \( \frac{1}{18} \).